Smoothed Particle Hydrodynamics

I present a review of Smoothed Particle Hydrodynamics (SPH), with the aim of providing a mathematically rigorous, clear derivation of the algorithms from first principles. The method of discretising a continuous field into particles using a smoothing kernel is considered, and also the errors associated with this approach. A fully conservative form of SPH is then derived from the Lagrangian, demonstrating the explicit conservation of mass, linear and angular momenta and energy/entropy. The method is then extended to self-consistently include spatially varying smoothing lengths, (self) gravity and various forms of artificial viscosity, required for the correct treatment of shocks. Finally two common methods of time integration are discussed, the Runge-Kutta-Fehlberg and leapfrog integrators, along with an overview of time-stepping criteria.

💡 Research Summary

This paper presents a thorough, first‑principles derivation and practical guide to Smoothed Particle Hydrodynamics (SPH), targeting both newcomers and seasoned practitioners. It begins by formalising the kernel‑based interpolation that underpins SPH: a smoothing kernel W(r,h) must be normalised, compactly supported, and symmetric, ensuring that any continuous field can be approximated to second‑order accuracy (O(h²)) when expressed as a weighted sum over particles. The authors analyse the discretisation error in detail, showing how particle spacing relative to the smoothing length h controls the magnitude of truncation errors.

From this foundation, the paper derives a fully conservative SPH formulation directly from a particle Lagrangian, L = ∑ₐ mₐ